Lambert summation

Last updated April 16, 2024

In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory.

Definition

Define the Lambert kernel by $L(x)=\log(1/x){\frac {x}{1-x}}$ with $L(1)=1$ . Note that $L(x^{n})>0$ is decreasing as a function of $n$ when $0<x<1$ . A sum $\sum _{n=0}^{\infty }a_{n}$ is Lambert summable to $A$ if $\lim _{x\to 1^{-}}\sum _{n=0}^{\infty }a_{n}L(x^{n})=A$ , written $\sum _{n=0}^{\infty }a_{n}=A\,\,(\mathrm {L} )$ .

Abelian and Tauberian theorem

Abelian theorem: If a series is convergent to $A$ then it is Lambert summable to $A$ .

Tauberian theorem: Suppose that $\sum _{n=1}^{\infty }a_{n}$ is Lambert summable to $A$ . Then it is Abel summable to $A$ . In particular, if $\sum _{n=0}^{\infty }a_{n}$ is Lambert summable to $A$ and $na_{n}\geq -C$ then $\sum _{n=0}^{\infty }a_{n}$ converges to $A$ .

The Tauberian theorem was first proven by G. H. Hardy and John Edensor Littlewood but was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation around the Lambert Tauberian theorem was resolved by Norbert Wiener.

Examples

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0\,(\mathrm {L} )$ , where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence ${\frac {\mu (n)}{n}}$ satisfies the Tauberian condition, therefore the Tauberian theorem implies $\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0$ in the ordinary sense. This is equivalent to the prime number theorem.
$\sum _{n=1}^{\infty }{\frac {\Lambda (n)-1}{n}}=-2\gamma \,\,(\mathrm {L} )$ where $\Lambda$ is von Mangoldt function and $\gamma$ is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to $-2\gamma$ . This is equivalent to $\psi (x)\sim x$ where $\psi$ is the second Chebyshev function.

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References

Jacob Korevaar (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. Vol. 329. Springer-Verlag. p. 18. ISBN 3-540-21058-X.
Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 159–160. ISBN 978-0-521-84903-6.
Norbert Wiener (1932). "Tauberian theorems". Ann. of Math. 33 (1). The Annals of Mathematics, Vol. 33, No. 1: 1–100. doi:10.2307/1968102. JSTOR 1968102.

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